With 3 states and 2 colors it turns out that with blank initial conditions all of the 2,985,984 possible Turing machines of this type quickly evolve to produce simple repetitive or nested behavior.

Both repetitive and nested behavior are seen to occur, though nothing more complicated is found.

In many cases, fairly regular repetitive or nested patterns are still produced.

If the elements of list correspond to values of a polynomial of degree n at successive integers, then Nest[d, list, n + 1] will contain only zeros.

When the paths do not cross themselves, nested structure is evident.

Strange attractors often have a nested structure with non-integer fractal dimension.

In most cases, this search will be done using some iterative scheme such as Newton's method; the result is that the boundaries between regions typically take on an intricate nested form. … But if an iterative scheme for minimization is used, these watersheds are typically no longer sharp, but take on a local nested structure, much as in picture (c) above.

As a approaches 3.569946, the period doublings get closer and closer together, and eventually a point is reached at which the sequence of leftmost digits is no longer repetitive but instead corresponds to the nested pattern formed after an infinite number of steps in the evolution of the substitution system. … When a is increased further, there is usually no longer repetitive or nested behavior.

The patterns that are obtained by this procedure turn out for reasons that are still not particularly clear to have a random but on average nested form. (Depending on precise details of the underlying model, very large clusters may sometimes not have nested forms, at least in 2D.)

In general, materials with nested structures do not necessarily yield discrete diffraction patterns.